Higher order representation stability and ordered configuration spaces of manifolds

TL;DR

This paper introduces secondary representation stability, a new pattern in the unstable homology of configuration spaces of manifolds, extending classical stability results using advanced algebraic and homological techniques.

Contribution

It defines and proves secondary representation stability for configuration spaces, providing new insights and extending previous stability results to nonorientable manifolds.

Findings

Secondary representation stability holds for rational homology of configuration spaces.

A new characterization of the homology of the complex of injective words is provided.

A new proof of integral representation stability for noncompact manifolds is established.

Abstract

Using the language of twisted skew-commutative algebras, we define \emph{secondary representation stability}, a stability pattern in the {\it unstable} homology of spaces that are representation stable in the sense of Church, Ellenberg, and Farb. We show that the rational homology of configuration spaces of ordered points in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing a point ``near infinity,'' secondary representation stability involves stabilizing by introducing a pair of orbiting points -- an operation that relates homology groups in different homological degrees. This result can be thought of as a representation-theoretic analogue of \emph{secondary homological stability} in the sense of Galatius, Kupers, and Randal-Williams. In the course of the proof we establish some additional results: we give a new characterization of the homology of the complex of injective words, and we give a new proof of integral representation stability for configuration spaces of noncompact manifolds, extending previous results to nonorientable manifolds.